Transcript Steven F. Ashby Center for Applied Scientific Computing

Data Mining
Classification: Basic Concepts, Decision
Trees, and Model Evaluation
Lecture Notes for Chapter 4
Introduction to Data Mining
by
Tan, Steinbach, Kumar
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
1
Classification: Definition

Given a collection of records (training set )
– Each record contains a set of attributes, one of the
attributes is the class.


Find a model for class attribute as a function
of the values of other attributes.
Goal: previously unseen records should be
assigned a class as accurately as possible.
– A test set is used to determine the accuracy of the
model. Usually, the given data set is divided into
training and test sets, with training set used to build
the model and test set used to validate it.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Illustrating Classification Task
Tid
Attrib1
Attrib2
Attrib3
1
Yes
Large
125K
No
2
No
Medium
100K
No
3
No
Small
70K
No
4
Yes
Medium
120K
No
5
No
Large
95K
Yes
6
No
Medium
60K
No
7
Yes
Large
220K
No
8
No
Small
85K
Yes
9
No
Medium
75K
No
10
No
Small
90K
Yes
Learning
algorithm
Class
Induction
Learn
Model
Model
10
Training Set
Tid
Attrib1
Attrib2
Attrib3
11
No
Small
55K
?
12
Yes
Medium
80K
?
13
Yes
Large
110K
?
14
No
Small
95K
?
15
No
Large
67K
?
Apply
Model
Class
Deduction
10
Test Set
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Examples of Classification Task

Predicting tumor cells as benign or malignant

Classifying credit card transactions
as legitimate or fraudulent

Classifying secondary structures of protein
as alpha-helix, beta-sheet, or random
coil

Categorizing news stories as finance,
weather, entertainment, sports, etc
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Classification Techniques
Decision Tree based Methods
 Rule-based Methods
 Memory based reasoning
 Neural Networks
 Naïve Bayes and Bayesian Belief Networks
 Support Vector Machines

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Example of a Decision Tree
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
Yes
6
No
Married
No
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
60K
Splitting Attributes
Refund
Yes
No
NO
MarSt
Single, Divorced
TaxInc
< 80K
NO
Married
NO
> 80K
YES
10
Model: Decision Tree
Training Data
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Another Example of Decision Tree
MarSt
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
Yes
6
No
Married
No
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
60K
Married
NO
Single,
Divorced
Refund
No
Yes
NO
TaxInc
< 80K
> 80K
NO
YES
There could be more than one tree that
fits the same data!
10
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Decision Tree Classification Task
Tid
Attrib1
Attrib2
Attrib3
1
Yes
Large
125K
No
2
No
Medium
100K
No
3
No
Small
70K
No
4
Yes
Medium
120K
No
5
No
Large
95K
Yes
6
No
Medium
60K
No
7
Yes
Large
220K
No
8
No
Small
85K
Yes
9
No
Medium
75K
No
10
No
Small
90K
Yes
Tree
Induction
algorithm
Class
Induction
Learn
Model
Model
10
Training Set
Tid
Attrib1
Attrib2
Attrib3
11
No
Small
55K
?
12
Yes
Medium
80K
?
13
Yes
Large
110K
?
14
No
Small
95K
?
15
No
Large
67K
?
Apply
Model
Class
Decision
Tree
Deduction
10
Test Set
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Apply Model to Test Data
Test Data
Start from the root of tree.
Refund
Yes
Refund Marital
Status
Taxable
Income Cheat
No
80K
Married
?
10
No
NO
MarSt
Single, Divorced
TaxInc
< 80K
NO
© Tan,Steinbach, Kumar
Married
NO
> 80K
YES
Introduction to Data Mining
4/18/2004
‹#›
Apply Model to Test Data
Test Data
Refund
Yes
Refund Marital
Status
Taxable
Income Cheat
No
80K
Married
?
10
No
NO
MarSt
Single, Divorced
TaxInc
< 80K
NO
© Tan,Steinbach, Kumar
Married
NO
> 80K
YES
Introduction to Data Mining
4/18/2004
‹#›
Apply Model to Test Data
Test Data
Refund
Yes
Refund Marital
Status
Taxable
Income Cheat
No
80K
Married
?
10
No
NO
MarSt
Single, Divorced
TaxInc
< 80K
NO
© Tan,Steinbach, Kumar
Married
NO
> 80K
YES
Introduction to Data Mining
4/18/2004
‹#›
Apply Model to Test Data
Test Data
Refund
Yes
Refund Marital
Status
Taxable
Income Cheat
No
80K
Married
?
10
No
NO
MarSt
Single, Divorced
TaxInc
< 80K
NO
© Tan,Steinbach, Kumar
Married
NO
> 80K
YES
Introduction to Data Mining
4/18/2004
‹#›
Apply Model to Test Data
Test Data
Refund
Yes
Refund Marital
Status
Taxable
Income Cheat
No
80K
Married
?
10
No
NO
MarSt
Single, Divorced
TaxInc
< 80K
NO
© Tan,Steinbach, Kumar
Married
NO
> 80K
YES
Introduction to Data Mining
4/18/2004
‹#›
Apply Model to Test Data
Test Data
Refund
Yes
Refund Marital
Status
Taxable
Income Cheat
No
80K
Married
?
10
No
NO
MarSt
Single, Divorced
TaxInc
< 80K
NO
© Tan,Steinbach, Kumar
Married
Assign Cheat to “No”
NO
> 80K
YES
Introduction to Data Mining
4/18/2004
‹#›
Decision Tree Classification Task
Tid
Attrib1
Attrib2
Attrib3
1
Yes
Large
125K
No
2
No
Medium
100K
No
3
No
Small
70K
No
4
Yes
Medium
120K
No
5
No
Large
95K
Yes
6
No
Medium
60K
No
7
Yes
Large
220K
No
8
No
Small
85K
Yes
9
No
Medium
75K
No
10
No
Small
90K
Yes
Tree
Induction
algorithm
Class
Induction
Learn
Model
Model
10
Training Set
Tid
Attrib1
Attrib2
Attrib3
11
No
Small
55K
?
12
Yes
Medium
80K
?
13
Yes
Large
110K
?
14
No
Small
95K
?
15
No
Large
67K
?
Apply
Model
Class
Decision
Tree
Deduction
10
Test Set
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Decision Tree Induction

Many Algorithms:
– Hunt’s Algorithm (one of the earliest)
– CART
– ID3, C4.5
– SLIQ,SPRINT
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
General Structure of Hunt’s Algorithm


Let Dt be the set of training records
that reach a node t
General Procedure:
– If Dt contains records that
belong the same class yt, then t
is a leaf node labeled as yt
– If Dt is an empty set, then t is a
leaf node labeled by the default
class, yd
– If Dt contains records that
belong to more than one class,
use an attribute test to split the
data into smaller subsets.
Recursively apply the
procedure to each subset.
© Tan,Steinbach, Kumar
Introduction to Data Mining
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
Yes
6
No
Married
No
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
60K
10
Dt
?
4/18/2004
‹#›
Hunt’s Algorithm
Don’t
Cheat
Refund
Yes
No
Don’t
Cheat
Don’t
Cheat
Refund
Refund
Yes
Yes
No
No
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
Yes
6
No
Married
No
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
60K
10
Don’t
Cheat
Don’t
Cheat
Marital
Status
Single,
Divorced
Cheat
Married
Single,
Divorced
Don’t
Cheat
© Tan,Steinbach, Kumar
Marital
Status
Married
Don’t
Cheat
Taxable
Income
< 80K
>= 80K
Don’t
Cheat
Cheat
Introduction to Data Mining
4/18/2004
‹#›
Tree Induction

Greedy strategy.
– Split the records based on an attribute test
that optimizes certain criterion.

Issues
– Determine how to split the records
How
to specify the attribute test condition?
How to determine the best split?
– Determine when to stop splitting
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Tree Induction

Greedy strategy.
– Split the records based on an attribute test
that optimizes certain criterion.

Issues
– Determine how to split the records
How
to specify the attribute test condition?
How to determine the best split?
– Determine when to stop splitting
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
How to Specify Test Condition?

Depends on attribute types
– Nominal
– Ordinal
– Continuous

Depends on number of ways to split
– 2-way split
– Multi-way split
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Splitting Based on Nominal Attributes

Multi-way split: Use as many partitions as distinct
values.
CarType
Family
Luxury
Sports

Binary split: Divides values into two subsets.
Need to find optimal partitioning.
{Sports,
Luxury}
CarType
© Tan,Steinbach, Kumar
{Family}
OR
Introduction to Data Mining
{Family,
Luxury}
CarType
{Sports}
4/18/2004
‹#›
Splitting Based on Ordinal Attributes

Multi-way split: Use as many partitions as distinct
values.
Size
Small
Large
Medium

Binary split: Divides values into two subsets.
Need to find optimal partitioning.
{Small,
Medium}

Size
{Large}
What about this split?
© Tan,Steinbach, Kumar
OR
{Small,
Large}
Introduction to Data Mining
{Medium,
Large}
Size
{Small}
Size
{Medium}
4/18/2004
‹#›
Splitting Based on Continuous Attributes

Different ways of handling
– Discretization to form an ordinal categorical
attribute
Static – discretize once at the beginning
 Dynamic – ranges can be found by equal interval
bucketing, equal frequency bucketing
(percentiles), or clustering.

– Binary Decision: (A < v) or (A  v)
consider all possible splits and finds the best cut
 can be more compute intensive

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Splitting Based on Continuous Attributes
Taxable
Income
> 80K?
Taxable
Income?
< 10K
Yes
> 80K
No
[10K,25K)
(i) Binary split
© Tan,Steinbach, Kumar
[25K,50K)
[50K,80K)
(ii) Multi-way split
Introduction to Data Mining
4/18/2004
‹#›
Tree Induction

Greedy strategy.
– Split the records based on an attribute test
that optimizes certain criterion.

Issues
– Determine how to split the records
How
to specify the attribute test condition?
How to determine the best split?
– Determine when to stop splitting
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
How to determine the Best Split
Before Splitting: 10 records of class 0,
10 records of class 1
Own
Car?
Yes
Car
Type?
No
Family
Student
ID?
Luxury
c1
Sports
C0: 6
C1: 4
C0: 4
C1: 6
C0: 1
C1: 3
C0: 8
C1: 0
C0: 1
C1: 7
C0: 1
C1: 0
...
c10
c11
C0: 1
C1: 0
C0: 0
C1: 1
c20
...
C0: 0
C1: 1
Which test condition is the best?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
How to determine the Best Split
Greedy approach:
– Nodes with homogeneous class distribution
are preferred
 Need a measure of node impurity:

C0: 5
C1: 5
C0: 9
C1: 1
Non-homogeneous,
Homogeneous,
High degree of impurity
Low degree of impurity
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Measures of Node Impurity

Gini Index

Entropy

Misclassification error
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
How to Find the Best Split
Before Splitting:
C0
C1
N00
N01
M0
A?
B?
Yes
No
Node N1
C0
C1
Node N2
N10
N11
C0
C1
N20
N21
M2
M1
Yes
No
Node N3
C0
C1
Node N4
N30
N31
C0
C1
M3
M12
N40
N41
M4
M34
Gain = M0 – M12 vs M0 – M34
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Measure of Impurity: GINI

Gini Index for a given node t :
GINI(t )  1  [ p( j | t )]2
j
(NOTE: p( j | t) is the relative frequency of class j at node t).
– Maximum (1 - 1/nc) when records are equally
distributed among all classes, implying least
interesting information
– Minimum (0.0) when all records belong to one class,
implying most interesting information
C1
C2
0
6
Gini=0.000
© Tan,Steinbach, Kumar
C1
C2
1
5
Gini=0.278
C1
C2
2
4
Gini=0.444
Introduction to Data Mining
C1
C2
3
3
Gini=0.500
4/18/2004
‹#›
Examples for computing GINI
GINI(t )  1  [ p( j | t )]2
j
C1
C2
0
6
P(C1) = 0/6 = 0
C1
C2
1
5
P(C1) = 1/6
C1
C2
2
4
P(C1) = 2/6
© Tan,Steinbach, Kumar
P(C2) = 6/6 = 1
Gini = 1 – P(C1)2 – P(C2)2 = 1 – 0 – 1 = 0
P(C2) = 5/6
Gini = 1 – (1/6)2 – (5/6)2 = 0.278
P(C2) = 4/6
Gini = 1 – (2/6)2 – (4/6)2 = 0.444
Introduction to Data Mining
4/18/2004
‹#›
Splitting Based on GINI


Used in CART, SLIQ, SPRINT.
When a node p is split into k partitions (children), the
quality of split is computed as,
k
GINIsplit
where,
© Tan,Steinbach, Kumar
ni
  GINI (i)
i 1 n
ni = number of records at child i,
n = number of records at node p.
Introduction to Data Mining
4/18/2004
‹#›
Binary Attributes: Computing GINI Index


Splits into two partitions
Effect of Weighing partitions:
– Larger and Purer Partitions are sought for.
Parent
B?
Yes
No
C1
6
C2
6
Gini = 0.500
Gini(N1)
= 1 – (5/7)2 – (2/7)2
= 0.408
Gini(N2)
= 1 – (1/5)2 – (4/5)2
= 0.320
© Tan,Steinbach, Kumar
Node N1
Node N2
C1
C2
N1
5
2
N2
1
4
Gini=0.371
Introduction to Data Mining
Gini(Children)
= 7/12 * 0.408 +
5/12 * 0.320
= 0.371
4/18/2004
‹#›
Categorical Attributes: Computing Gini Index


For each distinct value, gather counts for each class in
the dataset
Use the count matrix to make decisions
Multi-way split
Two-way split
(find best partition of values)
CarType
Family Sports Luxury
C1
C2
Gini
1
4
2
1
0.393
© Tan,Steinbach, Kumar
1
1
C1
C2
Gini
CarType
{Sports,
{Family}
Luxury}
3
1
2
4
0.400
Introduction to Data Mining
C1
C2
Gini
CarType
{Family,
{Sports}
Luxury}
2
2
1
5
0.419
4/18/2004
‹#›
Continuous Attributes: Computing Gini Index




Use Binary Decisions based on one
value
Several Choices for the splitting value
– Number of possible splitting values
= Number of distinct values
Each splitting value has a count matrix
associated with it
– Class counts in each of the
partitions, A < v and A  v
Simple method to choose best v
– For each v, scan the database to
gather count matrix and compute
its Gini index
– Computationally Inefficient!
Repetition of work.
© Tan,Steinbach, Kumar
Introduction to Data Mining
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
Yes
6
No
Married
No
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
60K
10
Taxable
Income
> 80K?
Yes
4/18/2004
No
‹#›
Continuous Attributes: Computing Gini Index...

For efficient computation: for each attribute,
– Sort the attribute on values
– Linearly scan these values, each time updating the count matrix
and computing gini index
– Choose the split position that has the least gini index
Cheat
No
No
No
Yes
Yes
Yes
No
No
No
No
100
120
125
220
Taxable Income
60
Sorted Values
70
55
Split Positions
75
65
85
72
90
80
95
87
92
97
110
122
172
230
<=
>
<=
>
<=
>
<=
>
<=
>
<=
>
<=
>
<=
>
<=
>
<=
>
<=
>
Yes
0
3
0
3
0
3
0
3
1
2
2
1
3
0
3
0
3
0
3
0
3
0
No
0
7
1
6
2
5
3
4
3
4
3
4
3
4
4
3
5
2
6
1
7
0
Gini
© Tan,Steinbach, Kumar
0.420
0.400
0.375
0.343
0.417
Introduction to Data Mining
0.400
0.300
0.343
0.375
0.400
4/18/2004
0.420
‹#›
Alternative Splitting Criteria based on INFO

Entropy at a given node t:
Entropy(t )   p( j | t ) log p( j | t )
j
(NOTE: p( j | t) is the relative frequency of class j at node t).
– Measures homogeneity of a node.
 Maximum
(log nc) when records are equally distributed
among all classes implying least information
 Minimum (0.0) when all records belong to one class,
implying most information
– Entropy based computations are similar to the
GINI index computations
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Examples for computing Entropy
Entropy(t )   p( j | t ) log p( j | t )
j
C1
C2
0
6
C1
C2
1
5
P(C1) = 1/6
C1
C2
2
4
P(C1) = 2/6
© Tan,Steinbach, Kumar
P(C1) = 0/6 = 0
2
P(C2) = 6/6 = 1
Entropy = – 0 log 0 – 1 log 1 = – 0 – 0 = 0
P(C2) = 5/6
Entropy = – (1/6) log2 (1/6) – (5/6) log2 (1/6) = 0.65
P(C2) = 4/6
Entropy = – (2/6) log2 (2/6) – (4/6) log2 (4/6) = 0.92
Introduction to Data Mining
4/18/2004
‹#›
Splitting Based on INFO...

Information Gain:
GAIN
n


 Entropy( p)    Entropy(i) 
 n

k
split
i
i 1
Parent Node, p is split into k partitions;
ni is number of records in partition i
– Measures Reduction in Entropy achieved because of
the split. Choose the split that achieves most reduction
(maximizes GAIN)
– Used in ID3 and C4.5
– Disadvantage: Tends to prefer splits that result in large
number of partitions, each being small but pure.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Splitting Based on INFO...

Gain Ratio:
GAIN
n
n
GainRATIO 
SplitINFO    log
SplitINFO
n
n
Split
split
k
i
i
i 1
Parent Node, p is split into k partitions
ni is the number of records in partition i
– Adjusts Information Gain by the entropy of the
partitioning (SplitINFO). Higher entropy partitioning
(large number of small partitions) is penalized!
– Used in C4.5
– Designed to overcome the disadvantage of Information
Gain
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Splitting Criteria based on Classification Error

Classification error at a node t :
Error (t )  1  max P(i | t )
i

Measures misclassification error made by a node.
 Maximum
(1 - 1/nc) when records are equally distributed
among all classes, implying least interesting information
 Minimum
(0.0) when all records belong to one class, implying
most interesting information
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Examples for Computing Error
Error (t )  1  max P(i | t )
i
C1
C2
0
6
C1
C2
1
5
P(C1) = 1/6
C1
C2
2
4
P(C1) = 2/6
© Tan,Steinbach, Kumar
P(C1) = 0/6 = 0
P(C2) = 6/6 = 1
Error = 1 – max (0, 1) = 1 – 1 = 0
P(C2) = 5/6
Error = 1 – max (1/6, 5/6) = 1 – 5/6 = 1/6
P(C2) = 4/6
Error = 1 – max (2/6, 4/6) = 1 – 4/6 = 1/3
Introduction to Data Mining
4/18/2004
‹#›
Comparison among Splitting Criteria
For a 2-class problem:
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Misclassification Error vs Gini
Parent
A?
Yes
No
Node N1
Gini(N1)
= 1 – (3/3)2 – (0/3)2
=0
Gini(N2)
= 1 – (4/7)2 – (3/7)2
= 0.489
© Tan,Steinbach, Kumar
Node N2
C1
C2
N1
3
0
N2
4
3
Gini=0.361
C1
7
C2
3
Gini = 0.42
Gini(Children)
= 3/10 * 0
+ 7/10 * 0.489
= 0.342
Gini improves !!
Introduction to Data Mining
4/18/2004
‹#›
Tree Induction

Greedy strategy.
– Split the records based on an attribute test
that optimizes certain criterion.

Issues
– Determine how to split the records
How
to specify the attribute test condition?
How to determine the best split?
– Determine when to stop splitting
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Stopping Criteria for Tree Induction

Stop expanding a node when all the records
belong to the same class

Stop expanding a node when all the records have
similar attribute values

Early termination (pre-pruning)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Decision Tree Based Classification

Advantages:
– Inexpensive to construct
– Extremely fast at classifying unknown records
– Easy to interpret for small-sized trees
– Accuracy is comparable to other classification
techniques for many simple data sets
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Example: C4.5
Simple depth-first construction.
 Uses Information Gain
 Sorts Continuous Attributes at each node.
 Needs entire data to fit in memory.
 Unsuitable for Large Datasets.
– Needs out-of-core sorting.

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Practical Issues of Classification

Underfitting and Overfitting

Missing Values

Costs of Classification
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Underfitting and Overfitting (Example)
500 circular and 500
triangular data points.
Circular points:
0.5  sqrt(x12+x22)  1
Triangular points:
sqrt(x12+x22) > 0.5 or
sqrt(x12+x22) < 1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Underfitting and Overfitting
Overfitting
Underfitting: when model is too simple, both training and test errors are large
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Overfitting due to Noise
Decision boundary is distorted by noise point
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Overfitting due to Insufficient Examples
Lack of data points in the lower half of the diagram makes it difficult
to predict correctly the class labels of that region
- Insufficient number of training records in the region causes the
decision tree to predict the test examples using other training
records that are irrelevant to the classification task
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Notes on Overfitting

Overfitting results in decision trees that are more
complex than necessary

Training error no longer provides a good estimate
of how well the tree will perform on previously
unseen records

Need new ways for estimating errors
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Estimating Generalization Errors



Re-substitution errors: error on training ( e(t) )
Generalization errors: error on testing ( e’(t))
Methods for estimating generalization errors:
– Optimistic approach: e’(t) = e(t)
– Pessimistic approach:



For each leaf node: e’(t) = (e(t)+0.5)
Total errors: e’(T) = e(T) + N  0.5 (N: number of leaf nodes)
For a tree with 30 leaf nodes and 10 errors on training
(out of 1000 instances):
Training error = 10/1000 = 1%
Generalization error = (10 + 300.5)/1000 = 2.5%
– Reduced error pruning (REP):

uses validation data set to estimate generalization
error
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Occam’s Razor

Given two models of similar generalization errors,
one should prefer the simpler model over the
more complex model

For complex models, there is a greater chance
that it was fitted accidentally by errors in data

Therefore, one should include model complexity
when evaluating a model
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Minimum Description Length (MDL)



X
X1
X2
X3
X4
y
1
0
0
1
…
…
Xn
1
A?
Yes
No
0
B?
B1
A
B2
C?
1
C1
C2
0
1
B
X
X1
X2
X3
X4
y
?
?
?
?
…
…
Xn
?
Cost(Model,Data) = Cost(Data|Model) + Cost(Model)
– Cost is the number of bits needed for encoding.
– Search for the least costly model.
Cost(Data|Model) encodes the misclassification errors.
Cost(Model) uses node encoding (number of children)
plus splitting condition encoding.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
How to Address Overfitting

Pre-Pruning (Early Stopping Rule)
– Stop the algorithm before it becomes a fully-grown tree
– Typical stopping conditions for a node:

Stop if all instances belong to the same class

Stop if all the attribute values are the same
– More restrictive conditions:
Stop if number of instances is less than some user-specified
threshold

Stop if class distribution of instances are independent of the
available features (e.g., using  2 test)


Stop if expanding the current node does not improve impurity
measures (e.g., Gini or information gain).
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
How to Address Overfitting…

Post-pruning
– Grow decision tree to its entirety
– Trim the nodes of the decision tree in a
bottom-up fashion
– If generalization error improves after trimming,
replace sub-tree by a leaf node.
– Class label of leaf node is determined from
majority class of instances in the sub-tree
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Example of Post-Pruning
Training Error (Before splitting) = 10/30
Class = Yes
20
Pessimistic error = (10 + 0.5)/30 = 10.5/30
Class = No
10
Training Error (After splitting) = 9/30
Pessimistic error (After splitting)
Error = 10/30
= (9 + 4  0.5)/30 = 11/30
PRUNE!
A?
A1
A4
A3
A2
Class = Yes
8
Class = Yes
3
Class = Yes
4
Class = Yes
5
Class = No
4
Class = No
4
Class = No
1
Class = No
1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Examples of Post-pruning
– Optimistic error?
Case 1:
Don’t prune for both cases
– Pessimistic error?
C0: 11
C1: 3
C0: 2
C1: 4
C0: 14
C1: 3
C0: 2
C1: 2
Don’t prune case 1, prune case 2
– Reduced error pruning?
Case 2:
Depends on validation set
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Handling Missing Attribute Values

Missing values affect decision tree construction in
three different ways:
– Affects how impurity measures are computed
– Affects how to distribute instance with missing
value to child nodes
– Affects how a test instance with missing value
is classified
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Computing Impurity Measure
Before Splitting:
Entropy(Parent)
= -0.3 log(0.3)-(0.7)log(0.7) = 0.8813
Tid Refund Marital
Status
Taxable
Income Class
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
Refund=Yes
Refund=No
5
No
Divorced 95K
Yes
Refund=?
6
No
Married
No
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
Entropy(Refund=Yes) = 0
9
No
Married
75K
No
10
?
Single
90K
Yes
Entropy(Refund=No)
= -(2/6)log(2/6) – (4/6)log(4/6) = 0.9183
60K
Class Class
= Yes = No
0
3
2
4
1
0
Split on Refund:
10
Missing
value
© Tan,Steinbach, Kumar
Entropy(Children)
= 0.3 (0) + 0.6 (0.9183) = 0.551
Gain = 0.9  (0.8813 – 0.551) = 0.3303
Introduction to Data Mining
4/18/2004
‹#›
Distribute Instances
Tid Refund Marital
Status
Taxable
Income Class
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
Yes
6
No
Married
No
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
60K
Tid Refund Marital
Status
Taxable
Income Class
10
90K
Single
?
Yes
10
Refund
Yes
No
Class=Yes
0 + 3/9
Class=Yes
2 + 6/9
Class=No
3
Class=No
4
Probability that Refund=Yes is 3/9
10
Refund
Yes
Probability that Refund=No is 6/9
No
Class=Yes
0
Cheat=Yes
2
Class=No
3
Cheat=No
4
© Tan,Steinbach, Kumar
Assign record to the left child with
weight = 3/9 and to the right child
with weight = 6/9
Introduction to Data Mining
4/18/2004
‹#›
Classify Instances
New record:
Married
Tid Refund Marital
Status
Taxable
Income Class
11
85K
No
?
Refund
NO
Divorced Total
Class=No
3
1
0
4
Class=Yes
0
1+6/9
1
2.67
Total
3
2.67
1
6.67
?
10
Yes
Single
No
Single,
Divorced
MarSt
Married
TaxInc
< 80K
NO
© Tan,Steinbach, Kumar
NO
Probability that Marital Status
= Married is 3/6.67
Probability that Marital Status
={Single,Divorced} is 3.67/6.67
> 80K
YES
Introduction to Data Mining
4/18/2004
‹#›
Search Strategy

Finding an optimal decision tree is NP-hard

The algorithm presented so far uses a greedy,
top-down, recursive partitioning strategy to
induce a reasonable solution

Other strategies?
– Bottom-up (CART)
– Bi-directional
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Decision Boundary
1
0.9
x < 0.43?
0.8
0.7
Yes
No
y
0.6
y < 0.33?
y < 0.47?
0.5
0.4
Yes
0.3
0.2
:4
:0
0.1
No
Yes
:0
:4
:0
:3
No
:4
:0
0
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
• Border line between two neighboring regions of different classes is
known as decision boundary
• Decision boundary is parallel to axes because test condition involves
a single attribute at-a-time
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Oblique Decision Trees
x+y<1
Class = +
Class =
• Test condition may involve multiple attributes
• More expressive representation
• Finding optimal test condition is computationally expensive
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Tree Replication
P
Q
S
0
R
0
Q
1
S
0
1
0
1
• Same subtree appears in multiple branches
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Model Evaluation

Metrics for Performance Evaluation
– How to evaluate the performance of a model?

Methods for Performance Evaluation
– How to obtain reliable estimates?

Methods for Model Comparison
– How to compare the relative performance
among competing models?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Model Evaluation

Metrics for Performance Evaluation
– How to evaluate the performance of a model?

Methods for Performance Evaluation
– How to obtain reliable estimates?

Methods for Model Comparison
– How to compare the relative performance
among competing models?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Metrics for Performance Evaluation
Focus on the predictive capability of a model
– Rather than how fast it takes to classify or
build models, scalability, etc.
 Confusion Matrix:

PREDICTED CLASS
Class=Yes
Class=Yes
ACTUAL
CLASS Class=No
© Tan,Steinbach, Kumar
a
c
Introduction to Data Mining
Class=No
b
d
a: TP (true positive)
b: FN (false negative)
c: FP (false positive)
d: TN (true negative)
4/18/2004
‹#›
Metrics for Performance Evaluation…
PREDICTED CLASS
Class=Yes
ACTUAL
CLASS

Class=No
Class=Yes
a
(TP)
b
(FN)
Class=No
c
(FP)
d
(TN)
Most widely-used metric:
ad
TP  TN
Accuracy 

a  b  c  d TP  TN  FP  FN
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Limitation of Accuracy

Consider a 2-class problem
– Number of Class 0 examples = 9990
– Number of Class 1 examples = 10

If model predicts everything to be class 0,
accuracy is 9990/10000 = 99.9 %
– Accuracy is misleading because model does
not detect any class 1 example
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Cost Matrix
PREDICTED CLASS
C(i|j)
Class=Yes
Class=Yes
C(Yes|Yes)
C(No|Yes)
C(Yes|No)
C(No|No)
ACTUAL
CLASS Class=No
Class=No
C(i|j): Cost of misclassifying class j example as class i
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Computing Cost of Classification
Cost
Matrix
PREDICTED CLASS
ACTUAL
CLASS
Model
M1
C(i|j)
+
-
+
-1
100
-
1
0
PREDICTED CLASS
ACTUAL
CLASS
+
-
+
150
40
-
60
250
Accuracy = 80%
Cost = 3910
© Tan,Steinbach, Kumar
Model
M2
ACTUAL
CLASS
PREDICTED CLASS
+
-
+
250
45
-
5
200
Accuracy = 90%
Cost = 4255
Introduction to Data Mining
4/18/2004
‹#›
Cost vs Accuracy
PREDICTED CLASS
Count
Class=Yes
Class=Yes
ACTUAL
CLASS
a
Class=No
Accuracy is proportional to cost if
1. C(Yes|No)=C(No|Yes) = q
2. C(Yes|Yes)=C(No|No) = p
b
N=a+b+c+d
Class=No
c
d
Accuracy = (a + d)/N
PREDICTED CLASS
Cost
Class=Yes
ACTUAL
CLASS
Class=No
Class=Yes
p
q
Class=No
q
p
© Tan,Steinbach, Kumar
Introduction to Data Mining
Cost = p (a + d) + q (b + c)
= p (a + d) + q (N – a – d)
= q N – (q – p)(a + d)
= N [q – (q-p)  Accuracy]
4/18/2004
‹#›
Cost-Sensitive Measures
a
P recision(p) 
ac
a
Recall (r) 
 TPRate
ab
2rp
2a
F - measure(F) 

r  p 2a  b  c



Precision is biased towards C(Yes|Yes) & C(Yes|No)
Recall is biased towards C(Yes|Yes) & C(No|Yes)
F-measure is biased towards all except C(No|No)
wa  w d
Weighted Accuracy 
wa  wb  wc  w d
1
© Tan,Steinbach, Kumar
Introduction to Data Mining
1
4
2
3
4
4/18/2004
‹#›
Model Evaluation

Metrics for Performance Evaluation
– How to evaluate the performance of a model?

Methods for Performance Evaluation
– How to obtain reliable estimates?

Methods for Model Comparison
– How to compare the relative performance
among competing models?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Methods for Performance Evaluation

How to obtain a reliable estimate of
performance?

Performance of a model may depend on other
factors besides the learning algorithm:
– Class distribution
– Cost of misclassification
– Size of training and test sets
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Learning Curve

Learning curve shows
how accuracy changes
with varying sample size

Requires a sampling
schedule for creating
learning curve:

Arithmetic sampling
(Langley, et al)

Geometric sampling
(Provost et al)
Effect of small sample size:
© Tan,Steinbach, Kumar
Introduction to Data Mining
-
Bias in the estimate
-
Variance of estimate
4/18/2004
‹#›
Methods of Estimation




Holdout
– Reserve 2/3 for training and 1/3 for testing
Random subsampling
– Repeated holdout
Cross validation
– Partition data into k disjoint subsets
– k-fold: train on k-1 partitions, test on the remaining one
– Leave-one-out: k=n
Bootstrap
– Sampling with replacement
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Model Evaluation

Metrics for Performance Evaluation
– How to evaluate the performance of a model?

Methods for Performance Evaluation
– How to obtain reliable estimates?

Methods for Model Comparison
– How to compare the relative performance
among competing models?
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
ROC (Receiver Operating Characteristic)
Developed in 1950s for signal detection theory to
analyze noisy signals
– Characterize the trade-off between positive
hits and false alarms
 ROC curve plots TPRate (on the y-axis) against
FPRate (on the x-axis)
 Performance of each classifier represented as a
point on the ROC curve
– changing the threshold of algorithm, sample
distribution or cost matrix changes the location
of the point

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
ROC Curve
(TPRate,FPRate):
 (0,0): declare everything
to be negative class
 (1,1): declare everything
to be positive class
 (1,0): ideal

Diagonal line:
– Random guessing
– Below diagonal line:
prediction is opposite of
the true class

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Using ROC for Model Comparison

No model consistently
outperform the other
 M1 is better for
small FPR
 M2 is better for
large FPR

Area Under the ROC
curve

Ideal:
 Area

Random guess:
 Area
© Tan,Steinbach, Kumar
Introduction to Data Mining
=1
= 0.5
4/18/2004
‹#›
How to Construct an ROC curve
Instance
P(+|A)
True Class
1
0.95
+
2
0.93
+
3
0.87
-
4
0.85
-
5
0.85
-
6
0.85
+
7
0.76
-
8
0.53
+
9
0.43
-
10
0.25
+
• Use classifier that produces
posterior probability for each
test instance P(+|A)
• Sort the instances according
to P(+|A) in decreasing order
• Apply threshold at each
unique value of P(+|A)
• Count the number of TP, FP,
TN, FN at each threshold
• TP rate, TPR = TP/(TP+FN)
• FP rate, FPR = FP/(FP + TN)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
How to construct an ROC curve
+
-
+
-
-
-
+
-
+
+
0.25
0.43
0.53
0.76
0.85
0.85
0.85
0.87
0.93
0.95
1.00
TP
5
4
4
3
3
3
3
2
2
1
0
FP
5
5
4
4
3
2
1
1
0
0
0
TN
0
0
1
1
2
3
4
4
5
5
5
FN
0
1
1
2
2
2
2
3
3
4
5
TPR
1
0.8
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.2
0
FPR
1
1
0.8
0.8
0.6
0.4
0.2
0.2
0
0
0
Class
P
Threshold
>=
ROC Curve:
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Rule-Based Classifier

Classify records by using a collection of
“if…then…” rules

Rule:
(Condition)  y
– where

Condition is a conjunctions of attributes

y is the class label
– LHS: rule antecedent or condition
– RHS: rule consequent
– Examples of classification rules:

(Blood Type=Warm)  (Lay Eggs=Yes)  Birds

(Taxable Income < 50K)  (Refund=Yes)  Evade=No
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Rule-based Classifier (Example)
Name
human
python
salmon
whale
frog
komodo
bat
pigeon
cat
leopard shark
turtle
penguin
porcupine
eel
salamander
gila monster
platypus
owl
dolphin
eagle
Blood Type
warm
cold
cold
warm
cold
cold
warm
warm
warm
cold
cold
warm
warm
cold
cold
cold
warm
warm
warm
warm
Give Birth
yes
no
no
yes
no
no
yes
no
yes
yes
no
no
yes
no
no
no
no
no
yes
no
Can Fly
no
no
no
no
no
no
yes
yes
no
no
no
no
no
no
no
no
no
yes
no
yes
Live in Water
no
no
yes
yes
sometimes
no
no
no
no
yes
sometimes
sometimes
no
yes
sometimes
no
no
no
yes
no
Class
mammals
reptiles
fishes
mammals
amphibians
reptiles
mammals
birds
mammals
fishes
reptiles
birds
mammals
fishes
amphibians
reptiles
mammals
birds
mammals
birds
R1: (Give Birth = no)  (Can Fly = yes)  Birds
R2: (Give Birth = no)  (Live in Water = yes)  Fishes
R3: (Give Birth = yes)  (Blood Type = warm)  Mammals
R4: (Give Birth = no)  (Can Fly = no)  Reptiles
R5: (Live in Water = sometimes)  Amphibians
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Application of Rule-Based Classifier

A rule r covers an instance x if the attributes of
the instance satisfy the condition of the rule
R1: (Give Birth = no)  (Can Fly = yes)  Birds
R2: (Give Birth = no)  (Live in Water = yes)  Fishes
R3: (Give Birth = yes)  (Blood Type = warm)  Mammals
R4: (Give Birth = no)  (Can Fly = no)  Reptiles
R5: (Live in Water = sometimes)  Amphibians
Name
hawk
grizzly bear
Blood Type
warm
warm
Give Birth
Can Fly
Live in Water
Class
no
yes
yes
no
no
no
?
?
The rule R1 covers a hawk => Bird
The rule R3 covers the grizzly bear => Mammal
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Rule Coverage and Accuracy
Tid Refund Marital
Status
Coverage of a rule:
1
Yes
Single
– Fraction of records
2
No
Married
that satisfy the
3
No
Single
antecedent of a rule
4
Yes
Married
5
No
Divorced
 Accuracy of a rule:
6
No
Married
– Fraction of records
7
Yes
Divorced
that satisfy both the
8
No
Single
9
No
Married
antecedent and
10 No
Single
consequent of a
(Status=Single)  No
rule

Taxable
Income Class
125K
No
100K
No
70K
No
120K
No
95K
Yes
60K
No
220K
No
85K
Yes
75K
No
90K
Yes
10
Coverage = 40%, Accuracy = 50%
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
How does Rule-based Classifier Work?
R1: (Give Birth = no)  (Can Fly = yes)  Birds
R2: (Give Birth = no)  (Live in Water = yes)  Fishes
R3: (Give Birth = yes)  (Blood Type = warm)  Mammals
R4: (Give Birth = no)  (Can Fly = no)  Reptiles
R5: (Live in Water = sometimes)  Amphibians
Name
lemur
turtle
dogfish shark
Blood Type
warm
cold
cold
Give Birth
Can Fly
Live in Water
Class
yes
no
yes
no
no
no
no
sometimes
yes
?
?
?
A lemur triggers rule R3, so it is classified as a mammal
A turtle triggers both R4 and R5
A dogfish shark triggers none of the rules
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Characteristics of Rule-Based Classifier

Mutually exclusive rules
– Classifier contains mutually exclusive rules if
the rules are independent of each other
– Every record is covered by at most one rule

Exhaustive rules
– Classifier has exhaustive coverage if it
accounts for every possible combination of
attribute values
– Each record is covered by at least one rule
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
From Decision Trees To Rules
Classification Rules
(Refund=Yes) ==> No
Refund
Yes
No
NO
Marita l
Status
{Single,
Divorced}
(Refund=No, Marital Status={Single,Divorced},
Taxable Income<80K) ==> No
{Married}
(Refund=No, Marital Status={Single,Divorced},
Taxable Income>80K) ==> Yes
(Refund=No, Marital Status={Married}) ==> No
NO
Taxable
Income
< 80K
NO
> 80K
YES
Rules are mutually exclusive and exhaustive
Rule set contains as much information as the
tree
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Rules Can Be Simplified
Tid Refund Marital
Status
Taxable
Income Cheat
1
Yes
Single
125K
No
2
No
Married
100K
No
3
No
Single
70K
No
4
Yes
Married
120K
No
5
No
Divorced 95K
6
No
Married
7
Yes
Divorced 220K
No
8
No
Single
85K
Yes
9
No
Married
75K
No
10
No
Single
90K
Yes
Refund
Yes
No
NO
{Single,
Divorced}
Marita l
Status
{Married}
NO
Taxable
Income
< 80K
NO
> 80K
YES
60K
Yes
No
10
Initial Rule:
(Refund=No)  (Status=Married)  No
Simplified Rule: (Status=Married)  No
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Effect of Rule Simplification

Rules are no longer mutually exclusive
– A record may trigger more than one rule
– Solution?
Ordered rule set
 Unordered rule set – use voting schemes


Rules are no longer exhaustive
– A record may not trigger any rules
– Solution?

Use a default class
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Ordered Rule Set

Rules are rank ordered according to their priority
– An ordered rule set is known as a decision list

When a test record is presented to the classifier
– It is assigned to the class label of the highest ranked rule it has
triggered
– If none of the rules fired, it is assigned to the default class
R1: (Give Birth = no)  (Can Fly = yes)  Birds
R2: (Give Birth = no)  (Live in Water = yes)  Fishes
R3: (Give Birth = yes)  (Blood Type = warm)  Mammals
R4: (Give Birth = no)  (Can Fly = no)  Reptiles
R5: (Live in Water = sometimes)  Amphibians
Name
turtle
© Tan,Steinbach, Kumar
Blood Type
cold
Give Birth
Can Fly
Live in Water
Class
no
no
sometimes
?
Introduction to Data Mining
4/18/2004
‹#›
Rule Ordering Schemes

Rule-based ordering
– Individual rules are ranked based on their quality

Class-based ordering
– Rules that belong to the same class appear together
Rule-based Ordering
Class-based Ordering
(Refund=Yes) ==> No
(Refund=Yes) ==> No
(Refund=No, Marital Status={Single,Divorced},
Taxable Income<80K) ==> No
(Refund=No, Marital Status={Single,Divorced},
Taxable Income<80K) ==> No
(Refund=No, Marital Status={Single,Divorced},
Taxable Income>80K) ==> Yes
(Refund=No, Marital Status={Married}) ==> No
(Refund=No, Marital Status={Married}) ==> No
© Tan,Steinbach, Kumar
(Refund=No, Marital Status={Single,Divorced},
Taxable Income>80K) ==> Yes
Introduction to Data Mining
4/18/2004
‹#›
Building Classification Rules

Direct Method:
Extract rules directly from data
 e.g.: RIPPER, CN2, Holte’s 1R


Indirect Method:
Extract rules from other classification models (e.g.
decision trees, neural networks, etc).
 e.g: C4.5rules

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Direct Method: Sequential Covering
1.
2.
3.
4.
Start from an empty rule
Grow a rule using the Learn-One-Rule function
Remove training records covered by the rule
Repeat Step (2) and (3) until stopping criterion
is met
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Example of Sequential Covering
(i) Original Data
© Tan,Steinbach, Kumar
(ii) Step 1
Introduction to Data Mining
4/18/2004
‹#›
Example of Sequential Covering…
R1
R1
R2
(iii) Step 2
© Tan,Steinbach, Kumar
(iv) Step 3
Introduction to Data Mining
4/18/2004
‹#›
Aspects of Sequential Covering

Rule Growing

Instance Elimination

Rule Evaluation

Stopping Criterion

Rule Pruning
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Rule Growing

Two common strategies
{}
Yes: 3
No: 4
Refund=No,
Status=Single,
Income=85K
(Class=Yes)
Refund=
No
Status =
Single
Status =
Divorced
Status =
Married
Yes: 3
No: 4
Yes: 2
No: 1
Yes: 1
No: 0
Yes: 0
No: 3
...
Income
> 80K
Yes: 3
No: 1
(a) General-to-specific
© Tan,Steinbach, Kumar
Introduction to Data Mining
Refund=No,
Status=Single,
Income=90K
(Class=Yes)
Refund=No,
Status = Single
(Class = Yes)
(b) Specific-to-general
4/18/2004
‹#›
Rule Growing (Examples)

CN2 Algorithm:
– Start from an empty conjunct: {}
– Add conjuncts that minimizes the entropy measure: {A}, {A,B}, …
– Determine the rule consequent by taking majority class of instances
covered by the rule

RIPPER Algorithm:
– Start from an empty rule: {} => class
– Add conjuncts that maximizes FOIL’s information gain measure:
R0: {} => class (initial rule)
 R1: {A} => class (rule after adding conjunct)
 Gain(R0, R1) = t [ log (p1/(p1+n1)) – log (p0/(p0 + n0)) ]
 where t: number of positive instances covered by both R0 and R1
p0: number of positive instances covered by R0
n0: number of negative instances covered by R0
p1: number of positive instances covered by R1
n1: number of negative instances covered by R1

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Instance Elimination

Why do we need to
eliminate instances?
R3
– Otherwise, the next rule is
identical to previous rule

Why do we remove
positive instances?
R1
+
class = +
+
– Ensure that the next rule is
different

Why do we remove
negative instances?
-
class = -
– Prevent underestimating
accuracy of rule
– Compare rules R2 and R3
in the diagram
© Tan,Steinbach, Kumar
Introduction to Data Mining
+ +
++
+
+
+
++
+ + +
+ +
-
-
-
+
+
+
+
+ +
-
-
-
+
+
+
-
+
+
+
+
-
-
R2
-
-
-
-
4/18/2004
‹#›
Rule Evaluation

Metrics:
– Accuracy
–
–
nc

n
nc  1
Laplace 
nk
nc  kp
M-estimate 
nk
© Tan,Steinbach, Kumar
Introduction to Data Mining
n : Number of instances
covered by rule
nc : Number of positive
instances covered by rule
k : Number of classes
p : Prior probability
4/18/2004
‹#›
Stopping Criterion and Rule Pruning

Stopping criterion
– Compute the gain
– If gain is not significant, discard the new rule

Rule Pruning
– Similar to post-pruning of decision trees
– Reduced Error Pruning:
Remove one of the conjuncts in the rule
 Compare error rate on validation set before and
after pruning
 If error improves, prune the conjunct

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Summary of Direct Method

Grow a single rule

Remove Instances from rule

Prune the rule (if necessary)

Add rule to Current Rule Set

Repeat
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Direct Method: RIPPER


For 2-class problem, choose one of the classes as
positive class, and the other as negative class
– Learn rules for positive class
– Negative class will be default class
For multi-class problem
– Order the classes according to increasing class
prevalence (fraction of instances that belong to a
particular class)
– Learn the rule set for smallest class first, treat the rest
as negative class
– Repeat with next smallest class as positive class
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Direct Method: RIPPER

Growing a rule:
– Start from empty rule
– Add conjuncts as long as they improve FOIL’s
information gain
– Stop when rule no longer covers negative examples
– Prune the rule immediately using incremental reduced
error pruning
– Measure for pruning: v = (p-n)/(p+n)
p: number of positive examples covered by the rule in
the validation set
 n: number of negative examples covered by the rule in
the validation set

– Pruning method: delete any final sequence of
conditions that maximizes v
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Direct Method: RIPPER

Building a Rule Set:
– Use sequential covering algorithm
Finds the best rule that covers the current set of
positive examples
 Eliminate both positive and negative examples
covered by the rule

– Each time a rule is added to the rule set,
compute the new description length
stop adding new rules when the new description
length is d bits longer than the smallest description
length obtained so far

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Direct Method: RIPPER

Optimize the rule set:
– For each rule r in the rule set R

Consider 2 alternative rules:
– Replacement rule (r*): grow new rule from scratch
– Revised rule(r’): add conjuncts to extend the rule r
Compare the rule set for r against the rule set for r*
and r’
 Choose rule set that minimizes MDL principle

– Repeat rule generation and rule optimization
for the remaining positive examples
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Indirect Methods
P
No
Yes
Q
No
-
Rule Set
R
Yes
No
Yes
+
+
Q
No
-
© Tan,Steinbach, Kumar
Yes
r1: (P=No,Q=No) ==> r2: (P=No,Q=Yes) ==> +
r3: (P=Yes,R=No) ==> +
r4: (P=Yes,R=Yes,Q=No) ==> r5: (P=Yes,R=Yes,Q=Yes) ==> +
+
Introduction to Data Mining
4/18/2004
‹#›
Indirect Method: C4.5rules
Extract rules from an unpruned decision tree
 For each rule, r: A  y,
– consider an alternative rule r’: A’  y where A’
is obtained by removing one of the conjuncts
in A
– Compare the pessimistic error rate for r
against all r’s
– Prune if one of the r’s has lower pessimistic
error rate
– Repeat until we can no longer improve
generalization error

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Indirect Method: C4.5rules

Instead of ordering the rules, order subsets of
rules (class ordering)
– Each subset is a collection of rules with the
same rule consequent (class)
– Compute description length of each subset
Description length = L(error) + g L(model)
 g is a parameter that takes into account the
presence of redundant attributes in a rule set
(default value = 0.5)

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Example
Name
human
python
salmon
whale
frog
komodo
bat
pigeon
cat
leopard shark
turtle
penguin
porcupine
eel
salamander
gila monster
platypus
owl
dolphin
eagle
© Tan,Steinbach, Kumar
Give Birth
yes
no
no
yes
no
no
yes
no
yes
yes
no
no
yes
no
no
no
no
no
yes
no
Lay Eggs
no
yes
yes
no
yes
yes
no
yes
no
no
yes
yes
no
yes
yes
yes
yes
yes
no
yes
Can Fly
no
no
no
no
no
no
yes
yes
no
no
no
no
no
no
no
no
no
yes
no
yes
Introduction to Data Mining
Live in Water Have Legs
no
no
yes
yes
sometimes
no
no
no
no
yes
sometimes
sometimes
no
yes
sometimes
no
no
no
yes
no
yes
no
no
no
yes
yes
yes
yes
yes
no
yes
yes
yes
no
yes
yes
yes
yes
no
yes
Class
mammals
reptiles
fishes
mammals
amphibians
reptiles
mammals
birds
mammals
fishes
reptiles
birds
mammals
fishes
amphibians
reptiles
mammals
birds
mammals
birds
4/18/2004
‹#›
C4.5 versus C4.5rules versus RIPPER
C4.5rules:
Give
Birth?
(Give Birth=No, Can Fly=Yes)  Birds
(Give Birth=No, Live in Water=Yes)  Fishes
No
Yes
(Give Birth=Yes)  Mammals
(Give Birth=No, Can Fly=No, Live in Water=No)  Reptiles
Live In
Water?
Mammals
Yes
( )  Amphibians
RIPPER:
No
(Live in Water=Yes)  Fishes
(Have Legs=No)  Reptiles
Sometimes
Fishes
Yes
Birds
© Tan,Steinbach, Kumar
(Give Birth=No, Can Fly=No, Live In Water=No)
 Reptiles
Can
Fly?
Amphibians
(Can Fly=Yes,Give Birth=No)  Birds
No
()  Mammals
Reptiles
Introduction to Data Mining
4/18/2004
‹#›
C4.5 versus C4.5rules versus RIPPER
C4.5 and C4.5rules:
PREDICTED CLASS
Amphibians Fishes Reptiles Birds
ACTUAL Amphibians
2
0
0
CLASS Fishes
0
2
0
Reptiles
1
0
3
Birds
1
0
0
Mammals
0
0
1
0
0
0
3
0
Mammals
0
1
0
0
6
0
0
0
2
0
Mammals
2
0
1
1
4
RIPPER:
PREDICTED CLASS
Amphibians Fishes Reptiles Birds
ACTUAL Amphibians
0
0
0
CLASS Fishes
0
3
0
Reptiles
0
0
3
Birds
0
0
1
Mammals
0
2
1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Advantages of Rule-Based Classifiers
As highly expressive as decision trees
 Easy to interpret
 Easy to generate
 Can classify new instances rapidly
 Performance comparable to decision trees

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›